For simplicity's sake I'm only talking about the US Presidential election, but you can apply the same logic to anything else
I think the biggest challenge to changing the US voting system (or any other nation-wide decision) is not some legal or logistical issue. I think its convincing the broader population that your solution is the better solution. Its not enough to just say 'I think this sucks, my solution is better and here is some math to prove it.' You need convince the majority of the people that this is both better and worth the transition cost.
That's not as simple as it sounds. There are downsides to Alternative Vote (some of which are described here, here and here) and Single Transferable Vote (look here, here and here) are not purely better than first past the post. Not to mention that the more rules and complications you add to your voting procedure the further way you get from a 'pure' democracy.
Plus there is a transition cost. I think its not unreasonable to say that the dollar cost for changing our voting infrastructure, professional training, machinenery and software, voter information and training, laws and regulations, etc is going to cost in the billions. Thats a lot of money, and its all going to be coming out of your taxes. You have to convince your fellow constituents that this is a worthy investment.